Non-commutative Tkachenko–Uspenskij conjecture

Let $X$ be a Tychonoff topological space, let $F_{\mathrm{SIN}}(X)$ be the free balanced topological group on $X$, and, for a Hilbert space ${\mathcal H}$, let $U({\mathcal H})_u$ denote its unitary group equipped with the uniform operator topology. Non-commutative Tkachenko–Uspenskij conjecture. Continuous homomorphisms

$$F_{\mathrm{SIN}}(X)\longrightarrow U({\mathcal H})_u$$

determine the topology of $F_{\mathrm{SIN}}(X)$. This is the proposed non-commutative, or quantized, analogue of the Tkachenko–Uspenskij theorem for free abelian topological groups; it was proved in the cited work.